Optimal experimental design for PSA-setting in gross alpha and gross beta measurements using LSC

The optimum pulse shape analysis (PSA) setting in gross alpha and gross beta measurements using liquid scintillation counting (LSC) is studied. Traditionally, an optimum setting is chosen such that the misclassification of alpha events equals that for beta events, i.e.,τ_α=τ_β. However, this might intuitively not be an optimum depending on the alpha-to-beta activity ratio (Norlin et al., 2023). For activity ratios far from one, say 1:100, i.e. when there is 100 times more beta activity compared to alpha activity and with misclassification of a few percent, more events from misclassified betas will enter the alpha window than true alpha events. This will have an impact on the uncertainty of the gross alpha activity and eventually the uncertainty will be too high to even be considered as detected, i.e. pass the detection criteria. In this work we derive an unbiased estimator for the activity vector, and an exact formula (without using uncertainty propagation approximation) for its covariance matrix, depending only on the mean and variance of the efficiency and misclassification factors (not the precise form of their distributions). We parametrize these means and variances with respect to the PSA value and obtain a PSA-dependent covariance matrix. We investigate optimal PSA values as a function of alpha-to-beta activity ratio for two optimal experiment designs based on this covariance matrix (A-optimality and E-optimality), and show that these optimal values differ slightly from each other, and differ markedly from the traditional PSA setting, especially for small alpha-beta ratios. Moreover, using the optimized rather than traditional PSA setting causes a dramatic decrease (orders of magnitude) in the relative standard uncertainty for both alpha and beta activity estimates, both for large and small activity ratios. The covariance matrix formula is derived from a general formula for the covariance of products of independent random matrices, a mathematical result of independent interest, not previously published to the best of our knowledge.